Determine if the lines are parallel, perpendicular, or neither. 10x + 8y = 16 and 5y = 4x - 15
Understand the Problem
The question is asking to determine the relationship between two lines given in equation form. Specifically, we need to analyze their slopes to see if they are parallel, perpendicular, or neither.
Answer
The lines are perpendicular.
Answer for screen readers
The lines are perpendicular.
Steps to Solve
- Identify the equations of the lines
The equations given are: $$ 10x + 8y = 16 $$ $$ 5y = 4x - 15 $$
- Convert the first equation to slope-intercept form
To find the slope, we need to rearrange the first equation into the form $y = mx + b$, where $m$ is the slope.
Starting with: $$ 10x + 8y = 16 $$
Subtract $10x$ from both sides: $$ 8y = -10x + 16 $$
Now, divide by 8: $$ y = -\frac{10}{8}x + 2 $$ $$ y = -\frac{5}{4}x + 2 $$
The slope of the first line ($m_1$) is $-\frac{5}{4}$.
- Convert the second equation to slope-intercept form
Starting with: $$ 5y = 4x - 15 $$
Divide by 5: $$ y = \frac{4}{5}x - 3 $$
The slope of the second line ($m_2$) is $\frac{4}{5}$.
- Analyze the slopes
To determine the relationship between the lines:
- Parallel lines have equal slopes ($m_1 = m_2$).
- Perpendicular lines have slopes that are negative reciprocals ($m_1 \cdot m_2 = -1$).
Now, calculate: $$ m_1 \cdot m_2 = -\frac{5}{4} \cdot \frac{4}{5} = -1 $$
This indicates that the lines are perpendicular.
The lines are perpendicular.
More Information
Perpendicular lines intersect at a right angle (90 degrees). The slopes of perpendicular lines multiply to give -1, confirming their relationship.
Tips
- Not converting to slope-intercept form: Always rearrange equations correctly to identify slopes.
- Miscalculating the product of the slopes: Ensure to perform multiplication carefully to check if they yield -1.
